New Fourier expansion for thermal buckling analysis of rectangular thin plates with various edge restraints

Abstract

For the first time, a modified two-dimensional Fourier series approach is proposed for new thermal buckling analysis of rectangular thin plates under various edge conditions. The solution form of plate deflection is considered to be in terms of a double Fourier Sine series (Navier-form solution) whose derivatives are determined via utilizing the Stoke’s transform technic. The present study exhibits the following significant merits: (a) the method adopted allows one to consider any possible combination of boundary conditions with no necessity to be satisfied in the Fourier series; (b) the given solution procedure is simple and straightforward since the complicated boundary value problem (BVP) of partial differential equation (PDE) can be changed into solving sets of linear algebra equations, which heavily decreases the complicated mathematical manipulations of plate thermal buckling problem; (c) all the results acquired converge rapidly because of using the sum function of series. Greeting agreements between the present analytical solutions with the numerical results provided by FEM testifies the accuracy of the approach proposed. The present results are believed to be severe as new benchmarks for validating other methods and providing better design for plate structures. The influences of the aspect ratio and boundary condition on the thermal buckling behaviors of plates are also investigated and discussed. Furthermore, it is capable to extend the present solution procedure to deal with problems of plates under more complex edge conditions by ways of utilizing other Fourier series.

Appendix

A: List of variables.

a, b, h	Length, width and thickness of the rectangular plate, respectively
$w\left( {x,y} \right)$	The deflection of a given point on plate surface (x, y)
$\mu ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} E,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} G_{xy} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} D$	The Poisson's ratio, Young’s moduli, shear modulus and flexural rigidity of the plate, respectively
$\sigma_{x} ,\sigma_{y}$	The normal stresses in the x and y directions, respectively
$\varepsilon_{x} ,\varepsilon_{y}$	The normal strains in the x and y directions, respectively
$\tau_{xy} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \gamma_{xy}$	The shear stress and strain in the x–y plane, respectively
$\chi$	Thermal expansion coefficients
$T_{0}$	The initial uniform temperature of the plate
$T$	The finial value of the temperature uniformly raised
$T_{ref}$	A stress-free reference temperature
$\Delta T = T - T_{0}$	An uniform temperature increment
$\Delta T_{cr}$	The critical buckling temperature of the plate
$N_{x}^{T} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} N_{y}^{T}$	Membrane force stimulated by the increasing of temperature
$M_{x} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} M_{y}$	Bending moments of the y and x axes, respectively
$- D \cdot F_{n} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - D \cdot G_{n}$	Fourier coefficients of the bending moment at edges x = 0, a, respectively
$- D \cdot I_{m} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - D \cdot H_{m}$	Fourier coefficients of the bending moment at edges y = 0,b, respectively
$w_{mn}$	The Fourier coefficient for the deflection of the plate
a/b	The aspect ratio of the plate
t	The number of terms of the series solution